Problem 3 - Precalculus

Question

Assume that the coordinates of the points \(P\) \(Q, R, S,\) and \(O\) are as follows: \(P(-1,3) \quad Q(4,6) \quad R(4,3) \quad S(5,9) \quad O(0,0)\) Draw the indicated vector (using graph paper) and compute its magnitude. Compute the sums using the definition. Use the parallelogram law to compute the sums. $$\overrightarrow{S Q}$$

Step-by-Step Solution

Answer

Vector \( \overrightarrow{SQ} = (-1, -3) \) with magnitude \( \sqrt{10} \).

Step 1: Understanding the Problem

We need to draw the vector \( \overrightarrow{SQ} \) from point \( S \) to point \( Q \), compute its magnitude, and then use the parallelogram law to verify the vector addition.

Step 2: Calculate Vector Components

To find the vector \( \overrightarrow{SQ} \), subtract the coordinates of point \( S(5,9) \) from the coordinates of point \( Q(4,6) \): \[ \overrightarrow{SQ} = (4 - 5, 6 - 9) = (-1, -3). \]

Step 3: Magnitude of the Vector

The magnitude of vector \( \overrightarrow{SQ} \) can be calculated using the formula for the magnitude of a vector: \[ |\overrightarrow{SQ}| = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}. \]

Step 4: Use Parallelogram Law

According to the parallelogram law, the sum of two vectors \( \overrightarrow{A} \) and \( \overrightarrow{B} \) is equal to the diagonal of the parallelogram formed, starting at the same initial point. In the case of a single vector \( \overrightarrow{SQ} \), it doesn't apply because there's no second vector, but the concept ensures vector placement and direction checks.

Step 5: Graph the Vector

On graph paper, plot point \( S(5,9) \) and point \( Q(4,6) \). Draw a directed line from \( S \) to \( Q \). The vector \( \overrightarrow{SQ} \) visually represents the change in position from \( S \) to \( Q \), matching our calculated direction and magnitude.

Key Concepts

Vector Addition

Vector addition is an essential concept in vector algebra. When we add two vectors, we combine them to get a resultant vector. This is crucial in physics for finding net forces, velocities, or other vector quantities.

To perform vector addition, we can use one of two primary methods: the triangle method or the parallelogram method. In the triangle method, we place the tail of the second vector at the head of the first vector. The resultant vector is then drawn from the tail of the first vector to the head of the second vector.

The parallelogram method involves placing both vectors such that they originate from the same point. Then, we complete the parallelogram and draw the diagonal from the common origin point; this diagonal represents the resultant vector.

Here are some key points about vector addition:

• Vectors can be added analytically by adding their corresponding components.
• The order of addition does not affect the resultant vector (commutative property).
• Vector addition is used for combining forces, velocities, and other vector quantities in physics and engineering.

Coordinate Geometry

Coordinate geometry helps us use algebraic techniques to understand geometric locations and shapes in a plane.

In our exercise, the coordinates of points such as \( S(5,9) \) and \( Q(4,6) \) are crucial in deriving vector components. By using their coordinates, we were able to determine the vector \( \overrightarrow{SQ} \) by performing simple arithmetic operations.

With coordinate geometry, we can find:

• Distances between points using the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
• Midpoints using the formula: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).
• Slopes of lines which are vital for understanding angles and intersections.

Coordinate geometry provides a bridge between algebra and traditional geometry allowing for deeper analytical insights.

Parallelogram Law

The parallelogram law is a geometric rule that aids in understanding vector addition. It gives us a visual way to determine the sum of two vectors. This concept involves drawing a parallelogram with each of the two vectors as adjacent sides.

The diagonal of the parallelogram from the common starting point is the vector sum. This visualization is particularly useful when analyzing forces in physics, as it helps in determining the resultant force acting on an object.

Consider these points when applying the parallelogram law:

• Both vectors should start from the same point for accurate representation.
• The "resultant vector" is calculated as the diagonal of the shape formed.
• This law verifies vector addition through spatial relationships rather than just arithmetic.

Although the problem at hand involves calculating a single vector \( \overrightarrow{SQ} \), understanding the parallelogram law ensures clarity on vector operations when multiple vectors interact.